Integral Representations for the Class of Generalized Metaplectic Operators

This article gives explicit integral formulas for the so-called generalized metaplectic operators, i.e. Fourier integral operators of Schrödinger type, having a symplectic matrix as their canonical transformation. These integrals are over specific linear subspaces of Rd , related to the d × d upper left-hand side submatrix of the underlying 2d×2d symplectic matrix. The arguments use the integral representations for the classical metaplectic operators obtained by Morsche and Oonincx in a previous paper, algebraic properties of symplectic matrices and time-frequency tools. As an application, we give a specific integral representation for solutions of the Cauchy problem of Schrödinger equations with bounded perturbations for every instant time t ∈ R, even at the (so-called) caustic points.